On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case

TL;DR

This paper analyzes the spectral properties of a 1D lattice spin-boson Hamiltonian, describing a two-level atom interacting with photons, establishing the essential spectrum location and conditions for finite or infinite discrete eigenvalues.

Contribution

It provides a detailed description of the essential spectrum and proves the finiteness of eigenvalues below it for any coupling, extending understanding of the lattice spin-boson model.

Findings

Location of the essential spectrum is characterized.

Finiteness of eigenvalues below the essential spectrum is proved for any coupling.

Conditions for the potential to have infinitely many eigenvalues are identified.

Abstract

A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model $H$ for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of $H,$ below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.